1. Unit 2 Lesson #3 Tangent Line Problems
Find the equations of tangents at given points
Find the points on the curve if tangent slope is known
Find equations of tangents parallel to given lines through a given point.
Find equations of tangents perpendicular to given lines through a given point.
Find equations of tangents to curves at
their point of intersection.

2. Unit 2 Lesson #3 Tangent Line Problems
Example 1: Find the equation of the tangent to
f (x) = x 2 + 2x – 3 at x = –1
Solution
Point of tangency is f (–1) = (–1)2 + 2(–1) – 3= – 4 → (– 1, – 4)
(–1, –4)
Slope of tangent = f ' (x) = 2x + 2
Slope of tangent at x = –1→ f ' (–1) = 2(–1)+ 2 = 0
– 4 = 0(–1) + b
b = – 4
y = – 4

3. Unit 2 Lesson #3 Tangent Line Problems
Example 2: Find the equation of the tangent to
f (x) = ½ x 4 + x – 2 at x = – 2
Unit 2 Lesson #3 Tangent Line Problems
Solution
Point of tangency is f (–2) = 4 → (–2, 4)
Slope of tangent f ' (x) = 4( ½) x 4 – 1 + 1(1)x 1 – 1 – 0
f ' (x) = 2x3 + 1
Slope of tangent at x = –2→ f '(–2) = 2(–2)3 + 1 = –15
–4 = –15(–2) + b
b = –26
y = – 15x – 26
(– 2 , 4)

4. Unit 2 Lesson #3 Tangent Line Problems
Example 3: Find the points where the graph of f(x) = x3 + 2x2 – 5x + 1 has
tangent lines with a slope of 2.
Unit 2 Lesson #3 Tangent Line Problems
f '(x) = 3x2 + 4x – 5
Solution
Derivative of the function gives the slope of the tangent lines
3x2 + 4x – 5 = 2
3x2 + 4x – 7 = 0
(3x + 7)(x – 1) = 0
x = − 𝟕 𝟑
𝒇 − 𝟕 𝟑 = − 𝟕 𝟑 𝟑 +𝟐 − 𝟕 𝟑 𝟐 −𝟓 − 𝟕 𝟑 +𝟏
or x = 1
𝒇 − 𝟕 𝟑 = 𝟐𝟗𝟑 𝟐𝟕
𝒇(𝟏) = (𝟏)𝟑 + 𝟐(𝟏)𝟐 – 𝟓(𝟏) + 𝟏 = −𝟏
The points on the graph of f (x) where the slope will be 2 are (− 𝟕 𝟑 , 𝟐𝟗𝟕 𝟐𝟕 ) and (1, – 1)

5. Unit 2 Lesson #3 Tangent Line Problems
Check by graphing
(− 𝟕 𝟑 , 𝟐𝟗𝟑 𝟐𝟕 )
(1, -1)

6. Unit 2 Lesson #3 Tangent Line Problems
Example 4
Given the curve g(x) = x3 – 12 x , at what points is the slope of the tangent line equal to 15?
g(x) = x3 – 12 x
g' (x) = 3x2 – 12
3x2 – 12 = 15 or
3x2 = 27
3x2 – 27 = 0
3(x2 – 9) = 0
x2 = 9
3(x – 3)(x + 3) = 0
x = 3 or – 3
Points are (3, – 9) and (– 3, 9)

7. Unit 2 Lesson #3 Tangent Line Problems
y = 15x + 54
m = 15
y = 15x – 54
m = 15
(-3, 9)
(3, -9)

8. Unit 2 Lesson #3 Tangent Line Problems
Example 5: Find the equation of a tangent to the graph of
f (x) = – 2x 2 that is parallel to y = – 4x – 5
Solution: The derivative of any function determines
the slope of the tangent line
f ' (x ) = – 4x and the slope of the given line is – 4
– 4 x = – 4
x = 1
f (x) = – 2x 2 → f (1) = – 2(1)2 = – 2
m = – 4 and P(1, – 2)
– 2 = – 4(1) + b
b = 2
(1, -2)
y = – 4 x + 2
y = 2x2
y = – 4x – 5

9. Unit 2 Lesson #3 Tangent Line Problems
Example 6
Find the equation of a tangent to the graph of f (x) = 3x2 – 4 that is perpendicular to 𝒚 =− 𝟏 𝟔 𝒙 +𝟏
Unit 2 Lesson #3 Tangent Line Problems
Solution: The derivative of any function determines the slope of the tangent line
f ' (x ) = 6x and the slope of the given line is − 1 6
so the slope of the perpendicular line is 6
6x = 6
x =1
f (x) = 3x 2 – 4 → f (1) = 3(1)2 – 4 = – 1
m = 6 and P(1, – 1)
– 1 = 6(1) + b
b = – 7
y = 6 x – 7

10. Unit 2 Lesson #3 Tangent Line Problems
Example 6 Continued
Unit 2 Lesson #3 Tangent Line Problems
Graph and check.
y = 6x – 7
𝒚=− 𝟏 𝟔 𝒙 + 𝟏
y = 3x2 – 4

11. Unit 2 Lesson #3 Tangent Line Problems
Example 7
Find the equation of the tangents to the curves
𝑓 (𝑥) = 1 3 𝑥 2 and 𝑔(𝑥) = 9 𝑥 at their point of intersection
Solution: Graphs intersect when
𝒙 𝟐 𝟑 = 𝟗 𝒙
x3 = 27
x = 3
𝒚= 𝟗 𝟑 =𝟑
Point of intersection is (3, 3)

12. Unit 2 Lesson #3 Tangent Line Problems
Example 7 Continued
𝒈 ′ 𝒙 =− 𝟗 𝒙 𝟐
𝒇 ′ 𝒙 = 𝟐 𝟑 𝒙
𝒇 ′(𝟑) = 𝟐 𝟑 (𝟑) = 𝟐
𝒈 ′ 𝟑 =− 𝟗 𝟑 𝟐 = −𝟏
Equations of Tangents
3 = 2(3) + b 3 = – 1(3) + b
b = – 3 b = 6
y = 2x – y = – x + 6

13. Unit 2 Lesson #3 Tangent Line Problems
Graph and Check
𝒇(𝒙) = 𝟏 𝟑 𝒙𝟐
(3, 3)
y = -x + 6
𝑔(𝑥)= 9 𝑥
y = 2x – 3

14. Unit 2 Lesson #3 Tangent Line Problems
Tangent Problems Assignment
Unit 2 Lesson #3 Tangent Line Problems
Complete Questions 1-14
Hand-in for marks